CR∞ membership for residually finite amenable groups of type FP∞

Determine whether every residually finite infinite amenable group of type FP∞ lies in CR∞, the class of groups whose trivial ZΓ-module admits projective resolutions satisfying the algebraic cheap rebuilding property in all degrees (i.e., for each T ≥ 1 and each residual chain, the coinvariant chain complexes admit uniform n-rebuildings with controlled ranks and norms for all n).

Background

The paper introduces algebraic cheap rebuilding properties CRn and CWRn for residually finite groups, designed to control (torsion) homology growth and admit combination theorems. The authors prove that residually finite infinite amenable groups of type FP∞ lie in CWR∞, which already implies vanishing torsion homology growth in all degrees and inclusion in H∞(F) for all fields.

However, CR∞ is a stronger, bootstrappable property with additional stability and structural consequences. Whether amenable groups of type FP∞ also satisfy the stronger CR∞ property remains unresolved.